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Theorem sbcom2 2189
 Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.)
Assertion
Ref Expression
sbcom2
Distinct variable groups:   ,   ,   ,

Proof of Theorem sbcom2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax6ev 1749 . 2
2 ax6ev 1749 . 2
3 2sb6 2188 . . . . . . . . . 10
4 alcom 1845 . . . . . . . . . 10
5 ancomst 452 . . . . . . . . . . 11
652albii 1641 . . . . . . . . . 10
73, 4, 63bitri 271 . . . . . . . . 9
8 2sb6 2188 . . . . . . . . 9
97, 8bitr4i 252 . . . . . . . 8
10 nfv 1707 . . . . . . . . 9
11 sbequ 2117 . . . . . . . . 9
1210, 11sbbid 2144 . . . . . . . 8
139, 12syl5bbr 259 . . . . . . 7
14 sbequ 2117 . . . . . . 7
1513, 14sylan9bb 699 . . . . . 6
16 nfv 1707 . . . . . . . 8
17 sbequ 2117 . . . . . . . 8
1816, 17sbbid 2144 . . . . . . 7
19 sbequ 2117 . . . . . . 7
2018, 19sylan9bbr 700 . . . . . 6
2115, 20bitr3d 255 . . . . 5
2221ex 434 . . . 4
2322exlimdv 1724 . . 3
2423exlimiv 1722 . 2
251, 2, 24mp2 9 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  [wsb 1739 This theorem is referenced by:  sbco4lem  2209  sbco4  2210  2mo  2373  2moOLD  2374  2eu6OLD  2384  cnvopab  5412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740
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